A note on splitting curves of plane quartics and multi-sections of rational elliptic surfaces (Q260603)

A reduced plane curve \(B\) of even degree \(2n\) gives a branched double cover of the projective plane \(\phi :S_{B} \to \mathbb{P}^{2}\). An irreducible plane curve \(C\) of degree \(d\) is simple contact curve of \(B\) if all the intersection points of \(B\) and \(C\) are smooth on both of \(B\) and \(C\) and the intersection multiplicity of each intersection point is \(2\). In the paper under review, the author gives a criterion of splitting for a simple contact curve of a reduced plane quartic curve \(B\) with at most simple double points (Proposition 3.3). When \(C_{1}\) and \(C_{2}\) are two splitting simple contact curve of B, the inverse image \(\phi ^{*}(C_{i})\) has two components \(C_{i} ^{\pm}\). The pair of intersection numbers \((C_{1}^{+} \cdot C_{2}^{+} , C_{1}^{+} \cdot C_{2}^{-})\) is called the splitting type of the triple \((C_{1}, C_{2};B)\). For a reduced plane quartic curve with at most simple double points, the splitting type of every possible two splitting simple contact curves can be calculated. (Proposition 3.4)NEWLINENEWLINE Using this result, the author constructs new examples of pairs of plane curves \(\mathcal{C}_{1}, \mathcal{C}_{2}\) such that the combinatorial types of \(\mathcal{C}_{1}\) and \(\mathcal{C}_{2}\) are equal but \((\mathbb{P}^{2}, \mathcal{C}_{1})\) is not homeomorphic to \((\mathbb{P}^{2}, \mathcal{C}_{2})\).